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a vague idea for f(f(x)) = exp(x)
#1
a vague idea ...


f(f(x)) = exp(x)

=> f(f(x)) + O(x) * f(x) = exp(x) with O(x) approaching 0.

=> f_n-1(f_n-1(x)) - exp(x) = - f_n(x)/ n!

=> - n! * ( f_n-1(f_n-1(x)) - exp(x) ) = f_n(x)

so starting at a suitable f_0(x) , whatever it may be , and applying the iteration :

- n! * ( f_n-1(f_n-1(x)) - exp(x) ) = f_n(x)


until it converges ( lim n -> oo ) at " f_oo(x) "

( convergeance depends upon f_0(x) of course )

if f_oo(x) is analytic ,

we have found f_oo(x) = f(x) with f(f(x)) =exp(x)


!!!!!!!!


i know : a vague idea.


but perhaps promising.


regards

tommy1729
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